Low-complexity non-data-aided estimation of symbol time offset in OFDM systems

ABSTRACT

A system and method involve receiving a plurality of samples of at least one orthogonal frequency division multiplex (OFDM) signal, the samples containing at least one complete OFDM symbol including data samples and a cyclic prefix comprising inter-symbol interference (ISI) samples and one or more ISI-free samples, and determining a symbol time offset estimate θ that minimizes the squared difference between the ISI-free samples and their corresponding data samples and satisfies a correlation based boundary condition.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. Non-Provisionalpatent application Ser. No. 14/091,048 filed Nov. 26, 2013, entitled“Non-Data-Aided Joint Time and Frequency Offset Estimate Method for OFDMSystems Using Channel Order Based Regression”, the content of which isfully incorporated by reference herein.

FEDERALLY-SPONSORED RESEARCH AND DEVELOPMENT

The Low-Complexity Non-Data-Aided Estimation of Symbol Time Offset inOFDM Systems is assigned to the United States Government and isavailable for licensing for commercial purposes. Licensing and technicalinquiries may be directed to the Office of Research and TechnicalApplications, Space and Naval Warfare Systems Center, Pacific, Code72120, San Diego, Calif., 92152; voice (619) 553-5118; email ssc_pacT2@navy.mil; reference Navy Case Number 103048.

BACKGROUND

Orthogonal frequency division multiplexing (OFDM) is a popularmulticarrier modulation method that has been adopted in numerouswireless networking and broadcasting standards such as IEEE 802.11a/g/p,LTE, and DVB-T/T2. Some of the key advantages of OFDM are its highspectral efficiency, robustness to inter-symbol interference (ISI)caused by multipath, and its ability to equalize wideband channels.However, OFDM is sensitive to time and frequency offsets, which need tobe estimated in order to correctly demodulate the received data.

Recent work has focused on blindly estimating these offsets.Non-data-aided estimators have the advantage of not requiring any knowntraining data, thus preserving high bandwidth efficiency. A majordrawback to many time offset estimators is that they are designed forsingle path channels and their estimation performance degrades inmultipath channels. Even estimators that are designed for multipathchannels tend to have high computational complexities.

A need exists for a new joint symbol time offset (STO) and carrierfrequency offset (CFO) estimator for OFDM systems when the channel orderis less than or equal to the length of the cyclic prefix that islow-complexity and does not suffer performance degradation in multipathchannels.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A shows a diagram of an OFDM signal containing a complete OFDMdata symbol.

FIG. 1B shows a diagram of the cyclic prefix portion of the OFDM datasymbol shown in FIG. 1.

FIG. 2 shows a graph illustrating the average of 1000 simulations of thecost functions versus symbol time offset estimates.

FIG. 3 shows a graph illustrating STO MSE estimation performance for3GPP Rax channel.

FIG. 4 shows a graph illustrating CFO MSE estimation performance for3GPP Rax channel.

FIG. 5 shows a graph illustrating STO MSE estimation performance for3GPP Tux channel.

FIG. 6 shows a graph illustrating CFO MSE estimation performance for3GPP Tux channel.

FIG. 7 shows a graph illustrating probability mass function of symboltiming error for a 3GPP Rax channel.

FIG. 8 shows a graph illustrating probability mass function of symboltiming error for a 3GPP Tux channel.

FIG. 9 shows a graph illustrating sensitivity of STO estimate to athreshold parameter ζ.

FIG. 10 shows a graph illustrating sensitivity of CFO estimate to athreshold parameter ζ.

FIG. 11 shows a diagram of an embodiment of an OFDM receiver system thatmay be used to implement embodiments of methods in accordance with theLow-Complexity Non-Data -Aided Estimation of Symbol Time Offset in OFDMSystems.

FIG. 12 shows a flowchart of an embodiment of a method in accordancewith the Low -Complexity Non-Data-Aided Estimation of Symbol Time Offsetin OFDM Systems.

DETAILED DESCRIPTION OF SOME EMBODIMENTS

The subject matter disclosed herein involves a joint STO and CFOestimator for OFDM systems when the channel order is less than or equalto the length of the cyclic prefix. The estimator is low-complexity anddoes not suffer performance degradation in multipath channels. Theestimator disclosed exploits the redundancy of the last sample of thereceived cyclic prefix, which is not corrupted by inter-symbolinterference (ISI), thus helping the estimator correctly estimate thetime of arrival in multipath channels.

The estimator discussed herein has been demonstrated to identify thecorrect STO with a higher probability than other estimators, which isimportant to OFDM systems so that pilots are not needed to refine theSTO estimate in the frequency domain and is critical to OFDM systemsusing differential modulation since pilots may not be available.

In general, the low-pass frequency-selective channel model is given by

$\begin{matrix}{{h(t)} = {\sum\limits_{l = 0}^{L - 1}{h_{l}{\delta\left( {t - {l\; T_{s}}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$where L is the order of the channel, h_(l) is the complex amplitude ofthe l-th multipath arrival, and T_(s) is the sampling period. Thechannel is assumed to stay unchanged over the duration of a couple ofOFDM symbols. The transmitted OFDM symbol s(n) n=0, . . . ,N+N_(cp)−1 isproduced by taking the N point inverse fast Fourier transform (IFFT) ofthe modulated data symbols {x_(d), d=0, . . . ,N−1} and pre-pending thelast N_(cp) samples. It is assumed that the channel order is less thanor equal to the length of the cyclic prefix (i.e., L≦N_(cp)). Hence, thecorrelation between the transmitted OFDM symbol's data portion and itscyclic prefix is given by

$\begin{matrix}{{E\left\lbrack {{s(a)}{s^{*}(b)}} \right\rbrack} = \left\{ \begin{matrix}{\sigma_{s}^{2},{{{if}\mspace{14mu} a} = b}} & {b \in \left\{ {0,\ldots\mspace{14mu},{N + N_{c\; p} - 1}} \right\}} \\{\sigma_{s}^{2},{{{if}\mspace{14mu} a} = {b + N}}} & {b \in \left\{ {0,\ldots\mspace{14mu},{N_{c\; p} - 1}} \right\}} \\{\sigma_{s}^{2},{{{if}\mspace{14mu} a} = {b - N}}} & {b \in \left\{ {N,\ldots\mspace{14mu},{N + N_{c\; p} - 1}} \right\}} \\{0,} & {otherwise}\end{matrix} \right.} & \left( {{Eq}.\mspace{14mu} 2} \right)\end{matrix}$where σ_(s) ² is the signal power. After convolution with the channel,the samples of the received OFDM symbol at the receiver are given by

$\begin{matrix}{{{r(k)} = {{{{\mathbb{e}}^{j\; 2{\pi ɛ}\;{k/N}}{\sum\limits_{l = 0}^{L - 1}{{h(l)}{s\left( {k - 1} \right)}}}} + {{n(k)}\mspace{14mu} k}} = \theta}},\ldots\mspace{14mu},{\theta + N + N_{c\; p} - 1}} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$where θε[0, N-1] is the integer STO, εε(−0.5,0.5] is the CFO normalizedto 1/NT_(s), and n is additive white Gaussian noise (AWGN) with varianceσ_(n) ². The received OFDM signal is assumed to be critically sampled(i.e. N+N_(cp) samples per OFDM symbol), and the STO θ is defined to bethe first arrival path received (i.e., the first sample of the receivedOFDM symbol still including the cyclic prefix). Due to the circularconvolution between the OFDM symbol and channel, the received cyclicprefix has L-1 samples corrupted by ISI and N_(cp)−L+1 ISI-free samples.

FIG. 1A shows a received OFDM signal 10 containing a complete OFDMsymbol 20. Symbol 20 includes a cyclic prefix 30 and a data portion 40.Cyclic prefix includes an inter -symbol interference (ISI) region 32 andan ISI-free region 34. Data portion 40 includes a first data portion 42,a second data portion 44, and a third data portion 46. Data portion 40contains samples representing the transmitted OFDM symbol. First dataportion 42 represents the samples that have not been affected byinterference. Second data portion 44 is a data region that correspondsto ISI region 32. Third data portion 46 is a data region thatcorresponds to ISI-free region 34. As shown in FIG. 1B, ISI region 32has a length of L-1 and ISI-free region 34 has a length of N_(cp)−L+1.

The proposed estimator disclosed herein exploits the assumption thatOFDM is based on, namely that the channel order is less than or equal tothe length of the cyclic prefix. This assumption guarantees that thereis always at least one ISI-free sample in the received cyclic prefix(i.e., r(θ+N_(cp)−1)). The embodiments disclosed herein aim to find theSTO estimate that minimizes the squared difference between this ISI freesample and its corresponding data sample (i.e., r(θ+N_(cp)+N −1)), whichcan be formulated from (Eq. 3) as

$\begin{matrix}{{\hat{\theta}}^{*} = {\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\min}{{{{\mathbb{e}}^{j\; 2{\pi ɛ}}{r\left( {\hat{\theta} + N_{cp} - 1} \right)}} - {r\left( {N + \hat{\theta} + N_{c\; p} - 1} \right)}}}^{2}}} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$Notice that the estimator is “shifted” so that r(θ+N_(cp)−1) is used inthe evaluation of the STO estimate at θ. In order to make the STOestimator's performance immune to the presence of CFO, the cost functionin (Eq. 4) is simplified to

$\begin{matrix}{{{r\left( {\hat{\theta} + N_{c\; p} - 1} \right)}}^{2} + {{r\left( {N + \hat{\theta} + N_{c\; p} - 1} \right)}}^{2} - {2\;{Re}\left\{ {{r\left( {\hat{\theta} + N_{cp} - 1} \right)}{r^{*}\left( {N + \hat{\theta} + N_{c\; p} - 1} \right)}{\mathbb{e}}^{j\; 2{\pi ɛ}}} \right\}}} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$Taking the magnitude of the last term in (Eq. 5), multiplying all theterms by −1, and accumulating over M OFDM symbols results in

$\begin{matrix}{{\hat{\theta}}^{*} = {{\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}} & \left( {{Eq}.\mspace{14mu} 6} \right)\end{matrix}$Let J denote the cost function in (Eq. 6). Since J utilizes thecorrelation between r(k) and r(k+N), its correlation is analyzed next.Using (Eq. 3), the correlation between the received samples r(k) andr(k+N) is

$\begin{matrix}{{E\left\lbrack {{r(k)}{r^{*}\left( {k + N} \right)}} \right\rbrack} = {E\left\lbrack {{{\mathbb{e}}^{{- j}\; 2{\pi ɛ}}{\sum\limits_{l_{1} = 0}^{L - 1}{{h\left( l_{1} \right)}{s\left( {k - l_{1}} \right)}{\sum\limits_{l_{2} = 0}^{L - 1}{{h^{*}\left( l_{2} \right)}{s^{*}\left( {k + N - l_{2}} \right)}}}}}} + {{\mathbb{e}}^{j\; 2{\pi ɛ}\;{k/N}}{\sum\limits_{l_{1} = 0}^{L - 1}{{h\left( l_{1} \right)}{s\left( {k - l_{1}} \right)}{n^{*}\left( {k + N} \right)}}}} + {{n(k)}{\mathbb{e}}^{{- j}\; 2{{{\pi ɛ}{({k + N})}}/N}}{\sum\limits_{l_{2} = 0}^{L - 1}{{h^{*}\left( l_{2} \right)}{s\left( {k + N - l_{2}} \right)}}}} + {{n(k)}{n^{*}\left( {k + N} \right)}}} \right\rbrack}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$All the terms in (Eq. 7), except the first one, are zero since E[n(k)]=0and n is independent of h and s. Using (Eq. 2), this simplifies (Eq. 7)to

$\begin{matrix}{{E\left\lbrack {{r(k)}{r^{*}\left( {k + N} \right)}} \right\rbrack} = {{E\left\lbrack {{{\mathbb{e}}^{{- {j2}}\;{\pi ɛ}}{\sum\limits_{l = 0}^{L - 1}{{h(l)}{h^{*}(l)}{s\left( {k - l} \right)}{s^{*}\left( {k + N - l} \right)}}}} + {{\mathbb{e}}^{{- j}\; 2{\pi ɛ}}\underset{l_{1} \neq l_{2}}{\Sigma\Sigma}{h\left( l_{1} \right)}{h^{*}\left( l_{2} \right)}{s\left( {k - l_{1}} \right)}{s^{*}\left( {k + N - l_{2}} \right)}}} \right\rbrack} = \mspace{79mu}\left\{ \begin{matrix}{{{\mathbb{e}}^{{- j}\; 2{\pi ɛ}}\sigma_{s}^{2}{\sum\limits_{l = 0}^{k - \theta}{{h(l)}{h^{*}(l)}}}},} & {k \in I_{1}} \\{{{\mathbb{e}}^{{- j}\; 2{\pi ɛ}}\sigma_{s}^{2}{\sum\limits_{l = 0}^{L - 1}{{h(l)}{h^{*}(l)}}}},} & {k \in I_{2}} \\{{{\mathbb{e}}^{{- j}\; 2{\pi ɛ}}\sigma_{s}^{2}{\sum\limits_{l = {k - N_{c\; p} + 1}}^{L - 1}{{h(l)}{h^{*}(l)}}}},} & {k \in I_{3}} \\{0,} & {otherwise}\end{matrix} \right.}} & \left( {{Eq}.\mspace{14mu} 8} \right)\end{matrix}$where I₁={θ, θ+1, . . . , θ+L−2}, I₂={θ+L−1, θ+L, . . . , θ+N_(cp)−1},and I₃={θ+N_(cp), θ+N_(cp)+1, . . . , θ+N_(cp)+L−2} since E[h(a)h*(b)]=0for a≠b. Note that the largest correlation values belong to I₂. It canalso be shown that the zero lag correlation is constant for all valuesof k, so it can be ignored in the analysis of J.

Recall that the purpose of the STO estimator is to correctly identify θ,so that the receiver can remove the cyclic prefix and subsequently takethe FFT of the data samples. Looking at (Eq. 8), if θ is not correctlyidentified by the STO estimator, there are still N_(cp)−L other points(L−1−N_(cp)+θ≦{circumflex over (θ)}<θ), upon which after cyclic prefixremoval, the data samples do not suffer from ISI. However, there wouldexist a phase offset that requires frequency domain pilots tocompensate. These N_(cp)−L points are not desirable for OFDM systemsusing differential modulation since frequency domain pilots may not beavailable. Thus, it is imperative that the STO estimator correctlyidentifies θ.

As used by the estimator embodiments disclosed herein, J is maximized byall {circumflex over (θ)} ε I₂. In order to avoid choosing one of theN_(cp)−L other points as the STO estimate, the boundary between thecorrelations belonging to I₂ and I₃ is used to assist the STO estimator(Eq. 6). Equation 8 shows that the difference in correlation between STOestimates θ+N_(cp)−1 and θ+N_(cp) is proportional to the product of thesignal power and power of the first channel tap. It has been observed inmeasurements of wireless channels that the power of each tap typicallydecreases with the delay, so the power of the first channel tap can beassumed to be large and thus provide a detectable boundary between I₂and I₃.

Hence, the following STO estimator is proposed:

$\begin{matrix}{{{\max\mspace{14mu}\hat{\theta}\mspace{14mu}{subject}\mspace{11mu}{to}\mspace{14mu}{\hat{\theta}}^{*}} = {\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{J\left( \hat{\theta} \right)}}}{{J\left( \hat{\theta} \right)} > {\zeta \times {J\left( {\hat{\theta}}^{*} \right)}}}} & \left( {{Eq}.\mspace{14mu} 9} \right)\end{matrix}$where ζ is a threshold parameter used to demarcate the boundary betweenthe correlations belonging to I₂ and the correlations belonging to I₃.To implement (Eq. 9), one would: 1) compute {circumflex over (θ)}* using(Eq. 6); 2) calculate the threshold ζ×J({circumflex over (θ)}*); and 3)find the largest STO estimate {circumflex over (θ)} whose cost functionJ lies above the threshold. Step 3 may be implemented as follows:starting with {circumflex over (θ)}* from Step 1, keep incrementing theSTO estimate by 1 sample until its corresponding cost function no longerlies above the threshold. The last STO estimate whose cost function liesabove the threshold is the STO estimate produced by (Eq. 9), which isdenoted by {circumflex over (θ)}**.

To help visualize the above disclosed estimator, FIG. 2 shows a graph200 illustrating the mean of 1,000 realizations of the cost function J(represented by line 110) versus STO estimates for a twelfth orderchannel, M=5, θ=30, and ζ=3. Step 1 produces {circumflex over (θ)}*=26,Step 2 produces the threshold line 120, and Step 3 produces {circumflexover (θ)}**=30 because the STO estimates {circumflex over (θ)}=27through {circumflex over (θ)}=30 all lie above the threshold whereas{circumflex over (θ)}=31 lies below threshold line 120.

The number of complex multiplications and additions performed is used tomeasure the computational complexity of the estimator. The proposedestimator performs 3N complex multiplications and 3N complex additionsfor each symbol used. For comparison, the computational complexity ofthe conditional maximum likelihood (CML) estimator discussed below (ablind estimator that has low computational complexity) is 3(N+N_(cp))−3complex multiplications and 6N+3N_(cp)−6 complex additions for eachsymbol used if three buffers of size N_(cp) are used.

Up to this point, the focus has been on STO estimation, however, OFDM issensitive to STO and CFO, both of which need to be estimated. The CFOestimator given by (Eq. 10) achieves the Cramér-Rao bound for AWGNchannels and performs well in multipath channels, and is thus adopted.Hence, the CFO estimate is given by

$\begin{matrix}{{\hat{ɛ}}^{**} = {{- \frac{1}{2\pi}}\angle{\sum\limits_{m = 0}^{M - 1}\left\{ {\sum\limits_{k = 0}^{N_{c\; p} - 1}{{r\left( {k + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {k + N + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)}}} \right\}}}} & \left( {{Eq}.\mspace{14mu} 10} \right)\end{matrix}$

FIGS. 3 through 6 show the STO and CFO mean squared error (MSE)performance of the estimator discussed herein and three other estimatorsthrough Monte Carlo simulations with 10,000 realizations, for M=5 andM=20. One estimator is discussed in a publication by Lopez -Salcedo etal., “Unified Framework for the Synchronization of Flexible MulticarrierCommunication Signals,” IEEE Trans. Signal Process., vol. 61, no. 4, pp.828-42, February 2013 (“CML estimator”), another estimator is discussedin a publication by Mo, R. et al., “A Joint Blind Timing and FrequencyOffset Estimator for OFDM Systems Over Frequency Selective FadingChannels,” IEEE Trans. Wireless Communications, vol. 5, no. 9, pp.2594-2604, September 2006 (“Mo estimator”), and the last estimator isdiscussed in a publication by Liu, X. et al., “Blind SymbolSynchronization for OFDM Systems in Multipath Fading Channels,” Proc.IEEE WiCOM, Chengdu, China, September 2010, pp. 1-4 (“Liu estimator”).

Generic OFDM signals are generated similar to the IEEE 802.11a signalstandard where N=64 and N_(cp)=16. In all of the simulations, θ=30,ε0.3, ζ=3, and BPSK modulation and Rayleigh fading channels are usedwhere the channel coefficients are normalized to unit power and followthe 3GPP Rural Area channel (Rax) and Typical Urban channel (Tux)models. By varying the bandwidth B of the generic OFDM signals, the Raxis effectively a twelfth order channel (i.e., B=20 MHz and L=12) and theTux is effectively a sixteenth order channel (i.e., B =7 MHz and L=16).

FIG. 3 shows a graph 200 illustrating STO MSE estimation performance for3GPP Rax channel. As shown, line 210 represents the performance of theestimator discussed herein for M =5, line 220 represents the performanceof the CML estimator for M=5, line 230 represents the performance of theMo estimator for M=5, line 240 represents the performance of the Liuestimator for M=5, line 250 represents the performance of the estimatordiscussed herein for M =20, line 260 represents the performance of theCML estimator for M=20, line 270 represents the performance of the Moestimator for M=20, and line 280 represents the performance of the Liuestimator for M=20.

FIG. 4 shows a graph 300 illustrating CFO MSE estimation performance for3GPP Rax channel. As shown, line 310 represents the performance of theestimator discussed herein for M =5, line 320 represents the performanceof the CML estimator for M=5, line 330 represents the performance of theMo estimator for M=5, line 340 represents the performance of theestimator discussed herein for M=20, line 350 represents the performanceof the CML estimator for M=20, and line 360 represents the performanceof the Mo estimator for M=20.

FIG. 5 shows a graph 400 illustrating STO MSE estimation performance for3GPP Tux channel. As shown, line 410 represents the performance of theestimator discussed herein for M =5, line 420 represents the performanceof the CML estimator for M=5, line 430 represents the performance of theMo estimator for M=5, line 440 represents the performance of the Liuestimator for M=5, line 450 represents the performance of the estimatordiscussed herein for M =20, line 460 represents the performance of theCML estimator for M=20, line 470 represents the performance of the Moestimator for M=20, and line 480 represents the performance of the Liuestimator for M=20.

FIG. 6 shows a graph 500 illustrating CFO MSE estimation performance for3GPP Tux channel. As shown, line 510 represents the performance of theestimator discussed herein for M =5, line 520 represents the performanceof the CML estimator for M=5, line 530 represents the performance of theMo estimator for M=5, line 540 represents the performance of theestimator discussed herein for M=20, line 550 represents the performanceof the CML estimator for M=20, and line 560 represents the performanceof the Mo estimator for M=20.

As shown in FIGS. 3-6, for low SNR, the CML estimator performs the best.However, the CML estimator exhibits a flooring effect for higher SNR dueto the finite cyclic prefix region used to evaluate the cost function.The estimator disclosed herein has sub-optimal performance for low SNRsince it only uses M samples to evaluate its cost function. However, forSNR>OdB, its performance improves as M increases. For SNR>3 dB and M=20,the estimator disclosed herein performs the best in terms of STO MSE anddoes not exhibit a floor. However, for SNR>4 dB and M=20, the estimatordisclosed herein does not perform as well as the CML estimator in termsof CFO MSE.

The reason for the superior STO estimation performance of the estimatordisclosed herein at high SNR can be explained by FIGS. 7 and 8. FIG. 7shows a graph 600 illustrating probability mass function of symboltiming error (defined to be {circumflex over (θ)}−θ) for a 3GPP Raxchannel, while FIG. 8 shows a graph 700 illustrating probability massfunction of symbol timing error for a 3GPP Tux channel. Results areshown for M=20 and E_(s)/N_(O)=15 dB. For the Rax and Tux channels, theproposed estimator disclosed herein identifies the correct STO almost92% and 88% of the time, respectively, which is over 29% higher than theMo estimator. Hence, for higher SNR, the proposed estimator performs thebest in terms of STO MSE because it is able to identify the correct STOwith a higher probability than the other estimators.

Finally, since the estimation performance of (Eq. 9) is dependent on ζ,which is dependent on the power of the first channel tap, thesensitivity of the estimator to ζ is explored. FIG. 9 shows a graph 800illustrating sensitivity of STO estimate to a threshold parameter ζ forthe proposed estimator, while FIG. 10 shows a graph 900 illustratingsensitivity of CFO estimate to ζ, each for M=20 and showing curvesrepresenting ζ values of 3, 5, 7, and 9. As shown, the estimatordisclosed herein is sensitive to ζ for SNR<10 dB, and less sensitive forSNR>10 dB, with smaller values of ζ performing the best. This makessense because, looking at FIG. 2, smaller values of ζ keep the thresholdhigh enough so that the boundary between the correlations belonging toI₂ and the correlations belonging to I₃ is correctly identified. Also,the estimator disclosed herein performs better in the Rax channel thanthe Tux channel because the power of the first channel tap is strongerin the Rax channel.

FIG. 11 shows a diagram of an embodiment of an OFDM receiver system 1000that may be used to implement the embodiments of the methods inaccordance with the Low-Complexity Non-Data-Aided Estimation of SymbolTime Offset in OFDM Systems. System 1000 may include antenna 1010 thatreceives an OFDM signal such as signal 10 shown in FIG. 1. The receivedsignal may then be converted to digital form using analog-to-digitalconverter (ADC) 1020. The output of ADC 1020 is sent to the STO/CFOestimation block 1030 and block 1040. The output block 1030 is a STO andCFO estimate, and based upon this estimate, the cyclic prefix of theOFDM symbol is removed at block 1040.

After removal of the cyclic prefix, the serial stream of OFDM symbolsare reshaped into N parallel streams 1050, upon which a Fast FourierTransform (FFT) is performed 1060. After the FFT 1060, a channelequalizer 1070 removes the channel's effect on the OFDM symbol in thefrequency domain. The output of channel equalizer 1070 is demodulated1080, where the OFDM symbols are converted into binary data, and the Nparallel streams of binary data are reshaped into one serial stream1090. The processing performed in blocks 1030-1090 may be performed by aprocessor that is connected to ADC 1020.

FIG. 12 shows a flowchart of an embodiment of a method 1100 inaccordance with the Low-Complexity Non-Data-Aided Estimation of SymbolTime Offset in OFDM Systems. As an example, method 1100 may be performedby system 1000 as shown in FIG. 11, using signal 10 as shown in FIG. 1,and will be discussed with reference thereto. Further, while FIG. 12shows one embodiment of method 1100 to include steps 1110-1140, otherembodiments of method 1100 may contain fewer or more steps. Further,while in some embodiments the steps of method 1100 may be performed asshown in FIG. 12, in other embodiments the steps may be performed in adifferent order, or certain steps may occur simultaneously with one ormore other steps.

Method 1100 may begin with step 1110, which involves receiving aplurality of samples of at least one transmitted OFDM signal, such assignal 10 shown in FIG. 1. As an example, the samples may be received atbox 1030 in FIG. 11. The samples contain at least one complete OFDMsymbol 20 including data samples in data portion 40 and a cyclic prefix30 comprising inter-symbol interference (ISI) samples in ISI region 32and one or more ISI-free samples in region 34.

Step 1120 involves determining a symbol time offset (STO) estimate θthat minimizes the squared difference between the ISI-free samples inISI region 32 and their corresponding data samples in third data portion46 and that also satisfies a correlation based boundary condition. Insome embodiments, step 1120 involves maximizing the set of possible STOestimates, {circumflex over (θ)}, subject to

${{\hat{\theta}}^{*} = {\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{J\left( \hat{\theta} \right)}}},$where N is the number of sub-carriers of the OFDM signal, ζ is athreshold parameter used to demarcate a boundary between correlationsbelonging to I₂ and correlations belonging to I₃, I₂={θ+L−1,θ+L, . . . ,θ+N_(cp)−1}, I₃={θ+N_(cp), θ+N_(cp)+1, . . . , θ+N_(cp)+L−2}, J is acost function, L is an order of a channel experienced by the ODFMsignal, N_(cp) is the length of the cyclic prefix, and L≦N_(cp).

In some embodiments, the step of maximizing {circumflex over (θ)}involves determining {circumflex over (θ)}*, calculating a thresholdζ×J({circumflex over (θ)}*), and determining {circumflex over (θ)}**,the largest STO estimate {circumflex over (θ)} whose cost function Jlies above the threshold. In some embodiments, determining {circumflexover (θ)}** involves incrementing, starting with {circumflex over (θ)}*,the STO estimate by one sample until its corresponding cost function nolonger lies above the threshold, wherein the last STO estimate whosecost function lies above the threshold is {circumflex over (θ)}**.

In some embodiments, {circumflex over (θ)}* is determined using theequation

$\begin{matrix}{{{\hat{\theta}}^{*} = {{\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}},} & \;\end{matrix}$where M is the total number of OFDM symbols, m is an indexing variable,r is the received OFDM samples, and r* is a complex conjugate of r.

In some embodiments, the cost function J is defined by the equation

${{J\left( \hat{\theta} \right)} = {{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}},$where M is the total number of OFDM symbols, m is an indexing variable,r is the received OFDM samples, and r* is a complex conjugate of r.

In some embodiments, method 1100 proceeds from step 1120 to step 1130,which involves determining a carrier frequency offset (CFO) using{circumflex over (θ)}**. In some embodiments, the CFO is determinedusing the equation

$\begin{matrix}{{{\hat{ɛ}}^{**} = {{- \frac{1}{2\pi}}\angle{\sum\limits_{m = 0}^{M - 1}\left\{ {\sum\limits_{k = 0}^{N_{c\; p} - 1}{{r\left( {k + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {k + N + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)}}} \right\}}}},} & \;\end{matrix}$where {circumflex over (ε)}** is the CFO, M is the total number of OFDMsymbols, m and k are indexing variables, r is the received OFDM samples,and r* is a complex conjugate of.

In some embodiments, method 1100 then proceeds to step 1140, whichinvolves using {circumflex over (θ)}** to remove the cyclic prefix fromthe OFDM symbol. In some embodiments, method 1100 proceeds to step 1140directly from step 1120.

Method 1100 may be implemented as a series of modules, eitherfunctioning alone or in concert, with physical electronic and computerhardware devices. Method 1100 may be computer-implemented as a programproduct comprising a plurality of such modules, which may be displayedfor a user.

Various storage media, such as magnetic computer disks, optical disks,and electronic memories, as well as non-transitory computer-readablestorage media and computer program products, can be prepared that cancontain information that can direct a device, such as a micro-controller, to implement the above-described systems and/or methods.Once an appropriate device has access to the information and programscontained on the storage media, the storage media can provide theinformation and programs to the device, enabling the device to performthe above-described systems and/or methods.

For example, if a computer disk containing appropriate materials, suchas a source file, an object file, or an executable file, were providedto a computer, the computer could receive the information, appropriatelyconfigure itself and perform the functions of the various systems andmethods outlined in the diagrams and flowcharts above to implement thevarious functions. That is, the computer could receive various portionsof information from the disk relating to different elements of theabove-described systems and/or methods, implement the individual systemsand/or methods, and coordinate the functions of the individual systemsand/or methods.

Many modifications and variations of the Low-Complexity Non-Data-AidedEstimation of Symbol Time Offset in OFDM Systems are possible in lightof the above description. Within the scope of the appended claims, theembodiments of the systems described herein may be practiced otherwisethan as specifically described. The scope of the claims is not limitedto the implementations and the embodiments disclosed herein, but extendsto other implementations and embodiments as may be contemplated by thosehaving ordinary skill in the art.

I claim:
 1. A method comprising the steps of: receiving, at an antennaof a receiver system, a plurality of samples of at least one orthogonalfrequency division multiplex (OFDM) signal, the samples containing atleast one complete OFDM symbol including data samples and a cyclicprefix comprising inter -symbol interference (ISI) samples and one ormore ISI-free samples; using an analog-to-digital converter (ADC)operatively connected to the antenna to convert the samples of the OFDMsignal into digital samples; using a processor operatively connected tothe ADC and using the digital samples to determine a symbol time offset(STO) estimate θ that minimizes the squared difference between theISI-free samples and their corresponding data samples and satisfies acorrelation based boundary condition; and using the processor to removethe cyclic prefix from the OFDM symbol based upon the determined STOestimate.
 2. The method of claim 1, wherein the step of determining theSTO estimate θ that minimizes the squared difference between theISI-free samples and their corresponding data samples and satisfies acorrelation based boundary condition comprises maximizing the set ofpossible symbol time offset (STO) estimates, {circumflex over (θ)},subject to${{\hat{\theta}}^{*} = {\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{J\left( \hat{\theta} \right)}}},$where N is the number of sub-carriers of the OFDM signal, ζ is athreshold parameter used to demarcate a boundary between correlationsbelonging to I₂ and correlations belonging to I₃, I₂={θ+L−1, θ+L, . . ., θ+N_(cp)−1}, I₃={θ+N_(cp), θ+N_(cp)+1, . . . , θ+N_(cp)+L−2}, J is acost function, L is an order of a channel experienced by the ODFMsignal, N_(cp) is the length of the cyclic prefix, and L≦N_(cp).
 3. Themethod of claim 2, wherein the step of maximizing {circumflex over (θ)}comprises the steps of: determining {circumflex over (θ)}*; calculatinga threshold ζ×J({circumflex over (θ)}*); and determining {circumflexover (θ)}**, the largest STO estimate {circumflex over (θ)} whose costfunction J lies above the threshold.
 4. The method of claim 3, whereinthe step of determining {circumflex over (θ)}** comprises the step ofincrementing, starting with {circumflex over (θ)}*, the STO estimate byone sample until its corresponding cost function no longer lies abovethe threshold, wherein the last STO estimate whose cost function liesabove the threshold is {circumflex over (θ)}**.
 5. The method of claim 4further comprising the step of determining a carrier frequency offset(CFO) using {circumflex over (θ)}**.
 6. The method of claim 5, whereinthe CFO is determined using the equation $\begin{matrix}{{{\hat{ɛ}}^{**} = {{- \frac{1}{2\pi}}\angle{\sum\limits_{m = 0}^{M - 1}\left\{ {\sum\limits_{k = 0}^{N_{c\; p} - 1}{{r\left( {k + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {k + N + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)}}} \right\}}}},} & \;\end{matrix}$ where {circumflex over (ε)}** is the CFO, M is the totalnumber of OFDM symbols, m and k are indexing variables, r is thereceived OFDM samples, and r* is a complex conjugate of r.
 7. The methodof claim 5, wherein the step of using the processor to remove the cyclicprefix from the OFDM symbol based upon the determined STO estimatecomprises using {circumflex over (θ)}** to remove the cyclic prefix fromthe OFDM symbol.
 8. The method of claim 3, wherein {circumflex over(θ)}* is determined using the equation $\begin{matrix}{{{\hat{\theta}}^{*} = {{\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}},} & \;\end{matrix}$ where M is the total number of OFDM symbols, m is anindexing variable, r is the received OFDM samples, and r* is a complexconjugate of r.
 9. The method of claim 3, wherein the cost function J isdefined by the equation${{J\left( \hat{\theta} \right)} = {{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}},$where M is the total number of OFDM symbols, m is an indexing variable,r is the received OFDM samples, and r* is a complex conjugate of r. 10.A system comprising: an antenna configured to receive a plurality ofsamples of at least one OFDM signal, the samples containing at least onecomplete OFDM symbol including a cyclic prefix comprising ISI samplesand one or more ISI-free samples; an analog-to-digital converter (ADC)operatively connected to the antenna, the ADC configured to convert thesamples of the OFDM signal into digital samples; and a processor,operatively connected to the ADC, configured to use the digital samplesto determine a symbol time offset (STO) estimate θ that minimizes thesquared difference between the ISI-free samples and their correspondingdata samples and satisfies a correlation based boundary condition, theprocessor further configured to remove the cyclic prefix from the OFDMsymbol based upon the determined STO estimate.
 11. The system of claim10, wherein the processor is configured to determine the STO estimate θthat minimizes the squared difference between the ISI-free samples andtheir corresponding data samples and satisfies a correlation basedboundary condition by maximizing the set of possible symbol time offset(STO) estimates, {circumflex over (θ)}, subject to${{\hat{\theta}}^{*} = {\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{J\left( \hat{\theta} \right)}}},$where N is the number of sub-carriers of the OFDM signal, ζ is athreshold parameter used to demarcate a boundary between correlationsbelonging to I₂ and correlations belonging to I₃, I₂={θ+L−1, θ+L, . . ., θ+N_(cp)−1}, I₃={θ+N_(cp), θ+N_(cp)+L−2}, J is a cost function, L isan order of a channel experienced by the ODFM signal, N_(cp) is thelength of the cyclic prefix, and L≦N_(cp).
 12. The system of claim 11,wherein the processor is configured to maximize {circumflex over (θ)} bydetermining {circumflex over (θ)}*, calculating a thresholdζ×J({circumflex over (θ)}*), and determining {circumflex over (θ)}**,the largest STO estimate {circumflex over (θ)} whose cost function Jlies above the threshold.
 13. The system of claim 12, wherein theprocessor is configured to determine {circumflex over (θ)}** byincrementing, starting with {circumflex over (θ)}*, the STO estimate byone sample until its corresponding cost function no longer lies abovethe threshold, wherein the last STO estimate whose cost function liesabove the threshold is {circumflex over (θ)}**.
 14. The system of claim13, wherein the processor is further configured to determine a carrierfrequency offset (CFO) using {circumflex over (θ)}**.
 15. The system ofclaim 14, wherein the processor is configured to determine the CFO usingthe equation $\begin{matrix}{{{\hat{ɛ}}^{**} = {{- \frac{1}{2\pi}}\angle{\sum\limits_{m = 0}^{M - 1}\left\{ {\sum\limits_{k = 0}^{N_{c\; p} - 1}{{r\left( {k + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {k + N + {\hat{\theta}}^{**} + {m\left( {N_{c\; p} + N} \right)}} \right)}}} \right\}}}},} & \;\end{matrix}$ where {circumflex over (ε)}** is the CFO, M is the totalnumber of OFDM symbols, m and k are indexing variables, r is thereceived OFDM samples, and r* is a complex conjugate of r.
 16. Thesystem of claim 14, wherein the processor is further configured to use{circumflex over (θ)}** to remove the cyclic prefix from the OFDMsymbol.
 17. The system of claim 12, wherein the processor is configuredto determine {circumflex over (θ)}* using the equation $\begin{matrix}{{{\hat{\theta}}^{*} = {{\underset{\hat{\theta} \in {\lbrack{0,{N - 1}}\rbrack}}{\arg\;\max}{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}},} & \;\end{matrix}$ where M is the total number of OFDM symbols, m is anindexing variable, r is the received OFDM samples, and r* is a complexconjugate of r.
 18. The system of claim 12, wherein the cost function Jis defined by the equation${{J\left( \hat{\theta} \right)} = {{{\sum\limits_{m = 0}^{M - 1}{2\;{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)} \times {r^{*}\left( {N + \hat{\theta} + \; N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {\hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}} - {\sum\limits_{m = 0}^{M - 1}{{r\left( {N + \hat{\theta} + N_{c\; p} - 1 + {m\left( {N_{c\; p} + N} \right)}} \right)}}^{2}}}},$where M is the total number of OFDM symbols, m is an indexing variable,r is the received OFDM samples, and r* is a complex conjugate of r.